On the uniqueness of the degree for nonlinear Fredholm maps of index zero between Banach manifolds

In some previous papers we presented a fairly simple construction of a topological degree for C^1 Fredholm maps of index zero between Banach manifolds which verifies the three fundamental properties of the classical degree theory: normalization, additivity and homotopy invariance. We show here that this degree is unique. Precisely, by an axiomatic approach similar to the one due to Amann-Weiss, we prove that there exists at most one real function satisfying the above properties, and this function must be integer valued.